Particle velocity estimation based on a two-microphone array and ...

Particle velocity estimation based on a two-microphone array and Kalman filter Mingsian R. Bai,a) Shen-Wei Juan, and Ching-Cheng Chen Department of Power Mechanical Engineering, National Tsing-Hua University, 101, Section 2, Kuang-Fu Road, Hsinchu 30013, Taiwan

(Received 10 July 2012; revised 30 November 2012; accepted 7 January 2013) A traditional method to measure particle velocity is based on the finite difference (FD) approximation of pressure gradient by using a pair of well matched pressure microphones. This approach is known to be sensitive to sensor noise and mismatch. Recently, a double hot-wire sensor termed Microflown became available in light of micro-electro-mechanical system technology. This sensor eliminates the robustness issue of the conventional FD-based methods. In this paper, an alternative two-microphone approach termed the u-sensor is developed from the perspective of robust adaptive filtering. With two ordinary microphones, the proposed u-sensor does not require novel fabrication technology. In the method, plane wave and spherical wave models are employed in the formulation of a Kalman filter with process and measurement noise taken into account. Both numerical and experimental investigations were undertaken to validate the proposed u-sensor technique. The results have shown that the proposed approach attained better performance than the FD method, C 2013 Acoustical Society of America. and comparable performance to a Microflown sensor. V [http://dx.doi.org/10.1121/1.4788986] PACS number(s): 43.38.Hz, 43.38.Ja, 43.60.Fg [DDE]

I. INTRODUCTION

Particle velocity is a key acoustical quantity relevant to many acoustic signal processing applications. For example, traditional sound intensity measurement is performed with the aid of a pair of well matched (in magnitude and phase) pressure microphones.1 For acoustic inverse reconstruction problems, particle velocity-based near-field acoustical holography, proved to yield better reconstruction quality than the conventional pressure-based methods.2,3 Recently, a hybrid array containing pressure and velocity sensors was implemented using maximum likelihood estimation for direction of arrival4 problems. In the emerging field of spatial audio reproduction, wave field synthesis renders a spatial sound field by approximating the Kirchhoff-Helmholtz integral equation, which requires an array of pressure and velocity sensors in the recording stage.5,6 Therefore, being able to access particle velocity could prove useful in many acoustical measurement and sensing scenarios. Despite the importance, particle velocity measurement is not as straightforward as pressure measurement. A traditional way of velocity measurement was based on the finite difference (FD)7,8 approximation of pressure gradient by using a pair of well matched pressure microphones. Since this approach is sensitive to sensor noise and mismatch, timeaveraging action has to be carried out to ensure the robustness. For instance, active sound intensity measured by an intensity probe can only be derived from the time-averaged (instead of instantaneous) spectral density functions. Recently, a double platinum hot-wire sensor termed Microflown9–11 became available in light of micro-electro-mechanical system (MEMS) technology. This sensor literally eliminates the robustness a)

Author to whom correspondence should be addressed. Electronic mail: [email protected]

J. Acoust. Soc. Am. 133 (3), March 2013

Pages: 1425–1432

issue inherent to the conventional FD-based methods. Wu and Wong12 have developed the three-dimensional source-localization method which combined a pressure-sensor and a velocity-sensor triad to estimate the pressure and velocity fields, respectively. The pressure-sensor can measure the received signal strength indication, whereas the velocitysensor triad can estimate the direction of arrival. Commercially available velocity sensors were used to estimate the direction of arrival12,13 in various environments. Whether the velocity sensor is a Microflown or a microphone pair is unknown to the authors. However, the localization method rather than the sensor was the main focus of the reference. The Monte Carlo method is used to simulate the signal and noise effects. The proposed method can accommodate prior known propagation loss. Wu et al.13 also examined the simple microphone array beamformers to enhance speech perception. Their approach does not require the prior information of interference and therefore is data independent. Three orthogonally oriented velocity sensors are employed to design the proposed “minimum-power distortionless response-diagonally loaded” (MPDR-DL) beamformer. Numerical simulation confirms that the MPDR-DL is capable of achieving beamforming performance as well as robustness against multiple simultaneous speakers. Notwithstanding the technological advances, is there any cost-effective alternative for measuring particle velocity? This paper addresses this question by revisiting the classical two-microphone approach from a Bayesian parameter estimation perspective.14 By assuming that the dynamic process and sensor noises are Gaussian, we estimate the particle velocity with a state observer based on the Kalman filter (KF).15,16 As an important assumption, for the KF technique to be effective requires that the direction and the location of the target source are known a priori. A frequency-domain

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state-space model17–19 is formulated in light of the equivalent source method (ESM).20–23 The processing noise and measurement noise models are incorporated into the formulation. The unknown source amplitudes of the virtual sources are taken as the state variables to estimate. Next, the particle velocity at the mid-point between the two microphones is calculated, based on the source amplitudes estimated previously. In addition, with inverse Fourier transform, the particle velocity and sound pressure can be used to calculate the instantaneous sound intensity. This paper is organized as follows. First, the FD approximation of the pressure gradient is reviewed, followed by the state-space equation based on the ESM. The spherical wave model and the plane wave model are introduced in the ESM. Next, formulation of particle velocity estimator based on KF observer is presented. Numerical simulations and experimental verifications are conducted to compare the FD approximation and the KF approach, with the Microflown sensor serving as the benchmark.

Let the time dependence be ejxt, and the line connecting two microphones be aligned with the x axis. In the planewave model, a plane wave is assumed to propagate in the positive x direction, as shown in Fig. 1(a). In phasor notation, the sound pressures picked up at the mth microphone (m ¼ 1, 2) are pm ðxm ; xÞ ¼ aðxÞejkxm ;

(3)

where k ¼ x/c is the wave number, x is the angular frequency, c is the speed of sound, xm is the x coordinate of the mth microphone (m ¼ 1,2), and a(x) is the unknown amplitude of the plane wave. The preceding equations can be written in the matrix form pm ðxÞ ¼ Gpl ðxÞaðxÞ;

(4)

where 

 ejkx1 Gpl ðxÞ ¼ jkx2 : e

(5)

II. THEORETICAL BACKGROUND

In this section, a brief review of particle velocity measurement based on the FD approximation of pressure gradient is given. Next, a state-space model is formulated on the basis of the ESM. Last, an KF-based velocity estimation procedure is presented.

Similarly, in the spherical-wave model, a spherical wave is assumed to be emitted by a monopole source located at xs, as shown in Fig. 1(b). Then, the sound pressures picked up at the mth microphone are

A. Finite difference approximation

where xm is the position vector of the mth microphone, xs is the position vector of the assumed monopole source position,

jxt

By the Euler’s equation with the time dependence e assumed, the particle velocity at the x direction can be written as ux ¼

j @p ; q0 x @x

pm ðxm ; xÞ ¼ qðxÞGðxm ; xs Þ;

m ¼ 1; 2

(6)

(1)

pffiffiffiffiffiffiffi where p is sound pressure, j ¼ 1, q0 is air density, and x denotes angular frequency. Thus, the particle velocity can be obtained by evaluating the pressure gradient @[email protected] By using the two-point finite difference (FD) method, particle velocity is readily approximated as ux ¼

j @p j p2  p1  ; q0 x @x q0 x Dx

(2)

where p1 and p2 are pressure measurements at two adjacent points and Dx is the spacing between two pressure microphones. Despite the simplicity, the FD method suffers from robustness issue against sensor noise and mismatch. B. Equivalent source models for the two-microphone array

In this work, we will pose the velocity estimation problem from a viewpoint different from the Microflown and FD approach. We shall establish the acoustic system by using the ESM by which the sound field of interest is represented with an array of simple sources.20–23 Two simple source models are introduced, including the plane-wave model and the spherical-wave model. 1426

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FIG. 1. Equivalent source models. (a) Plane-wave model, (b) sphericalwave model. Bai et al.: Particle velocity sensor

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q(x) is the unknown source strength of the monopole, and G(xm, xs) is the free-space Green’s function defined as Gðxm ; xs Þ ¼

jq0 xejkrm ; 4prm

(7)

with q0 is the mass density of the medium and rm ¼ kxm  xs k being the distance between the monopole and the mth microphone. In matrix form, this can be written as pm ðxÞ ¼ Gsp ðxÞqðxÞ;

(8)

where r22 is the variance of measurement noise in each channel, I is the identity matrix. Unlike the process equation, how to choose the state transition equation A in the process equation is not as straightforward. One solution to this problem is to impose a “smoothness” condition to the state variables. For the planewave model, it is assumed that the source amplitude satisfies the smoothness condition described by the following linear first-order difference equation (with frequency x suppressed for simplicity): aðn þ 1Þ  aðnÞ ¼ v1 ðnÞ:

where 2

3 ejkr1 jq x 6 r1 7 7: Gsp ¼ 0 6 4p 4 ejkr2 5 r2

(9)

Similarly, for the spherical-wave model, it is assumed that the source strength satisfies the smoothness condition described by qðn þ 1Þ  qðnÞ ¼ v1 ðnÞ:

Therefore, the velocity estimation problem boils down to the estimation of the unknown amplitude (in the plane-wave case) and the unknown source strength (in the monopole case). To this end, a model-based approach is proposed in light of state-space formulation. C. State-space formulation based on the ESM

The method proposed in this paper is based on the statespace representation17–19 which is a versatile mathematical model for dynamic systems. It consists of a linear system of difference equations xðn þ 1Þ ¼ AxðnÞ þ v1 ðnÞ;

(10)

yðnÞ ¼ CxðnÞ þ v2 ðnÞ;

(11)

(13)

(14)

Both models lead to the simple choice of the state transition matrix A ¼ 1. The scalar process noise term v1 is assumed to be a zero-mean, wide-sense stationary (WSS), i.i.d., and Gaussian white noise defined by the correlation  2 r1 ; n ¼ k H (15) E½v1 ðnÞv1 ðkÞ ¼ 0; n 6¼ k; where r21 is the variance of process noise. The noise terms v1 and v2 are assumed to be statistically uncorrelated. In summary, the state-space equations are written as aðn þ 1Þ ¼ aðnÞ þ v1 ðnÞ; pm ðnÞ ¼ Gpl aðnÞ þ v2 ðnÞ

(16)

for the plane-wave model and where x is the state vector, A is the state transition matrix, and v1 is the process noise vector, y is the output vector, C is the output matrix, and v2 is the measurement noise vector. Note that n is the frame index since the acoustical problem in the paper is formulated in the frequency domain. The equations are termed the process equation and the measurement equation, respectively. The preceding spherical-wave model and plane-wave model can be cast into the state-space formalism with proper realizations. In the first place, the ESM formulations in Eq. (8) can be readily used as the measurement equation in Eq. (11) by choosing the unknown source amplitude or strength as the state variable to estimate. Specifically, for the plane-wave model, we let x ¼ aðxÞ 2 C1 , y ¼ pm 2 C2 , C ¼ Gpl 2 C21 , whereas for the spherical-wave model, we let x ¼ qðxÞ 2 C1 , y ¼ pm 2 C2 , C ¼ Gsp 2 C21 . The measurement noise vector v2 is a zero-mean, wide-sense stationary (WSS), independent and identically distributed (i.i.d.), and Gaussian white noise defined by the correlation matrix. Note that both models are single-input-two-output systems because there is only one scalar source parameter to estimate in both cases. E½v2 ðnÞvH 2 ðkÞ ¼



r22 I;

n¼k

0;

n 6¼ k;

J. Acoust. Soc. Am., Vol. 133, No. 3, March 2013

(12)

qðn þ 1Þ ¼ qðnÞ þ v1 ðnÞ; pm ðnÞ ¼ Gsp qðnÞ þ v2 ðnÞ

(17)

for the spherical-wave model, respectively. D. Velocity estimation based on the frequency-domain Kalman filter

The velocity estimation problem is essentially a parameter estimation problem. A comprehensive exposition on parameter estimation theory has been given in the textbook by Kay.14 According to Kay, there are two categories of estimation methods—the classical approaches and the Bayesian approaches. The main difference between these two approaches lies in that the classical approach assumes the parameters to be deterministic but unknown constants, while the Bayesian approach treats them to be stochastic and random variables. The Bayesian approach incorporates the prior knowledge into the estimation process. In general, introduction of prior information improves the estimation accuracy. One of the approaches to implement Bayesian inference is the KF, which is a widely used adaptive filter well suited to linear and Gaussian estimation problems. Given the dynamic system represented by the state equation, a standard Bai et al.: Particle velocity sensor

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procedure of KF involving two key steps of prediction and update can be applied. To save space, instead of reviewing the KF algorithm in detail, we simply refer the interested readers to the relevant literature, e.g., Haykin’s text.15 We summarize the complete process of particle velocity estimation using the two-microphone array in conjunction with the KF in a flowchart illustrated in Fig. 2. The timedomain sound data acquired at two pressure microphones are first transformed to the frequency domain, with the aid of fast Fourier transform (FFT). Next, a source model has to be selected. The state-space equation is constructed, depending on whether the plane-wave model or the spherical-wave model is selected. Then, state estimation is performed by using the frequency-domain KF16 to calculate the source amplitude or strength. The index n denotes the FFT frame index, which renders the adaptive filtering procedure framebased processing. With the source amplitude or strength estimated, we may proceed with computing the sound pressure and particle velocity at the array center as follows. For the plane-wave model, pðxÞ ¼ aðxÞejkxc ; 1 pðxÞ uðxÞ ¼ aðxÞejkxc ¼ ; q0 c q0 c

(18)

with xc being the coordinate of the array center and u(x) being the Fourier transform of the particle velocity in the x direction. For the spherical-wave model, ejkrc ; pðxÞ ¼ jq0 xqðxÞ 4prc   jkrc   1 e j pðxÞ ; ¼ 1 uðxÞ ¼ qðxÞ þ jk 4prc r kr q0 c

(19)

with rc being the distance between the assumed monopole source and the array center [Fig. 1(b)] and u(x) being the Fourier transform of the particle velocity in the radial direction. Last, inverse fast Fourier transform is utilized to convert the frequency-domain pressure and particle velocity data into the time-domain data, p(t) and u(t). With the time-domain data, we can readily calculate the instantaneous acoustic intensity by IðtÞ ¼ pðtÞuðtÞ:

(20)

III. NUMERICAL AND EXPERIMENTAL VERIFICATIONS

In order to validate the proposed two-microphone u-sensor, numerical and experimental investigations were undertaken. In the numerical simulation, relative velocity reconstruction error was employed as the performance metric: EðxÞ ¼

juðxÞ  ur ðxÞj  100%; juðxÞj

(21)

where jj symbolizes the absolute value and x denotes the frequency, and u and ur denote the desired and the reconstructed particle velocities, respectively. The first simulation compares the FD method and the KF method based on the plane-wave model. White noise signal with 5.79 m3/s (rms) was employed as the source waveform. Two microphones were spaced by 5 mm. The sampling rate was 8 kHz and the FFT size was 512. Figure 3

FIG. 2. Flowchart of particle velocity estimation algorithm using the KF. 1428

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FIG. 3. (Color online) The relative estimation error (%) of particle velocity. The source input is clean signal. The KF approach is based on the plane wave model. Bai et al.: Particle velocity sensor

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shows the relative errors of particle velocity estimated by the FD method and the KF method with clean source signal. The estimation errors obtained using both methods are within 1% with two approaches. Next, white noises uncorrelated with the preceding clean signal were injected with 40 and 20 dB signal-to-noise ratios (SNRs). The results were shown in Figs. 4(a) and 4(b). As expected, the relative velocity errors estimated by both methods were larger than the previous case of clean signal. With 40 and 20 dB SNRs, the velocity error of KF approach is smaller than the FD approach throughout the frequency range, as shown in Fig. 4. Velocity reconstruction error of the FD approach is very large, especially at low frequencies. Situation got even worse for the case of 20 dB SNR, the relative velocity error increases markedly in the frequency range 50 Hz  2.5 kHz. The KF approach had yielded significantly better estimation than the FD method. In addition, another simulation was conducted to examine the effects due to microphone mismatch on the velocity estimation. Consider two microphones with 5% mismatch in

FIG. 4. (Color online) The relative estimation errors (%) of particle velocity with (a) 40 dB SNR and (b) 20 dB SNR. The KF approach is based on the plane wave model. J. Acoust. Soc. Am., Vol. 133, No. 3, March 2013

FIG. 5. (Color online) The relative estimation errors (%) of particle velocity with 5% magnitude and 5 phase mismatch.

magnitude and 5 mismatch in phase. The results in Fig. 5 exhibit sharp contrast of estimation performance using two approaches. The low-frequency velocity errors had reached radically high 560% at 200 Hz (not shown in the figure) for the FD approach, while the velocity errors had still dwelled in the range 5%–6% for the KF approach. Notably, the FD approach was extremely sensitive to sensor mismatch, especially at low frequencies. Overall, the KF approach was clearly more robust in velocity estimation than the FD approach in terms of sensor noise and mismatch. Next, the second simulation was conducted to compare the FD method and the KF based on the spherical-wave model. Again, white noise signal with 5.79 m3/s (rms) was employed as the source waveform. The distance between the monopole source and the center of microphone array was 0.6 m. The FFT size was 512. Two microphones were spaced by 5 mm with 8 kHz sampling rate, as in the plane-wave case. Figure 6 shows the relative errors of particle velocity estimated by the FD method and the KF method with clean

FIG. 6. (Color online) The relative estimation error (%) of particle velocity. The source input is clean signal. The KF approach is based on the spherical wave model. Bai et al.: Particle velocity sensor

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source signal. All velocity estimation errors were less than 1% with two approaches. Next, white noises uncorrelated with the preceding clean signal were injected with 40 and 20 dB SNRs. The results were shown in Figs. 7(a) and 7(b). As expected, the relative velocity errors estimated by both methods were larger than the previous case of clean signal. For the noise-corrupted signal with 40 and 20 dB SNRs, a marked decrease in the velocity errors can be seen in 700 Hz4 kHz (Fig. 7). As in the plane-wave model, the KF approach performed significantly better than the FD approach. In addition, a numerical simulation was conducted to examine the effects due to microphone mismatch on the velocity estimation. Consider the microphones with 5% mismatch in magnitude and 5 mismatch in phase. The results were shown in Fig. 8. The low-frequency velocity errors had reached radically high 511% at 200 Hz (not shown in the figure) for the FD approach, while the velocity errors remained in the range 5%–6% for the KF approach. The FD approach was extremely sensitive to sensor mismatch, especially in low frequencies. Suffice it to say that the KF approach was

FIG. 8. (Color online) The relative estimation errors (%) of particle velocity with 5% magnitude and 5 phase mismatch.

more robust than the FD approach in terms of sensor noise and mismatch. The spherical-wave model had yielded slightly better estimation than the plane-wave model at most frequencies. Next, we calculated the error of the time-domain data estimated using the KF approach based on the sphericalwave model. The test signal was white noise with 5.79 m3/s (rms). The relative velocity estimation error was defined as Et ¼

kz  zr k2  100%; kzk2

(22)

where z and zr denote the desired and the reconstructed data vectors of the time-domain sound pressure, particle velocity, or sound intensity. The results were summarized in Table I. The metric of relative velocity error is applied to evaluate the performance of the FD method. In Table I, the time-domain estimation errors of sound pressure, particle velocity, and sound intensity were within the range 0%–2%, 0%–68%, and 0%–42%. In addition to the simulation above, experimental investigations were conducted to further justify the proposed usensor. The FD and KF algorithms were implemented on the platform of the LabVIEW and PXI system from National InstrumentsV. The sampling rate was 16 kHz. To avoid aliasing, a low-pass filter with the cutoff frequency 7.5 kHz was used. A 2-s white noise clip was utilized as the input signal to a loudspeaker. Two commercial MEMS microphones R

TABLE I. The relative time-domain estimation errors of sound pressure, particle velocity, and sound intensity. Method

FIG. 7. (Color online) The relative estimation errors (%) of particle velocity with (a) 40 dB SNR and (b) 20 dB SNR. The KF approach is based on the spherical wave model. 1430

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Test signal

Sound pressure Particle velocity Sound intensity

KF

Clean signal SNR ¼ 40 dB SNR ¼ 20 dB

0.00% 0.15% 1.47%

0.00% 0.17% 1.70%

0.00% 0.26% 2.77%

FD

Clean signal SNR ¼ 40 dB SNR ¼ 20 dB

1.10% 1.11% 1.79%

0.37% 6.83% 68.20%

1.34% 4.28% 41.50%

Bai et al.: Particle velocity sensor

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FIG. 9. (Color online) The experimental arrangement. (a) Experimental setup, (b) close-up view of the two-microphone u-sensor, (c) close-up view of the Microflown.

(Knowles SPM0410HR5H-PB) were employed to construct the u-sensor, as shown in Fig. 9. The physical sizes of twomicrophone and u-sensor Microflown are defined as follows: (1) Two-microphone u-sensor: 15 mm (W)  12 mm (L), as shown in Fig. 9(b); (2) Microflown: diameter: 12.7 mm (D)  130 mm (L), as shown in Fig. 9(c). The proposed u-sensor is substantially shorter than the Microflown sensor. The microphones were spaced by 5 mm. The distance between the u-sensor and the source loudspeaker was set to be 0.6 m. To benchmark the proposed usensor, a Microflown sensor was used in the experiment. The particle velocity spectra obtained using four approaches including the FD, the KF based on the plane-wave model, the KF based on the spherical-wave model, and the Microflown were compared in Fig. 10. In the experiment, the desired velocity was measured by a Microflown sensor. The

FIG. 10. (Color online) Particle velocity estimated or measured by using the FD, KF, and Microflown approaches. J. Acoust. Soc. Am., Vol. 133, No. 3, March 2013

relative velocity estimation errors of the FD method, the KF method based on the plane-wave model and the KF method based on the spherical-wave model were 18.90%, 9.10%, and 9.17%. Clearly, the results estimated using the four approaches followed similar trend below 4 kHz. Unlike the other three methods, the particle velocity spectrum obtained using the FD method rolled off at 4 kHz. The result obtained using the plane-wave model was nearly identical to that obtained using the spherical-wave model. While the KF methods had yielded comparable estimation with that of Microflown in the frequency range 500 Hz to 7.5 kHz, they outperformed the FD method. IV. CONCLUSIONS

In this work, a two-microphone u-sensor has been developed for particle velocity estimation. Plane-wave and spherical-wave models served as the equivalent source models. Velocity estimation was carried out, with the aid of the Kalman filter. Numerical and experimental validations have been undertaken. The results have shown that, for clean signals, the FD method yielded comparable velocity estimation error with the KF method. However, for noise-corrupted signals, the KF method significantly outperformed the FD method in velocity estimation. In addition, the KF-based u-sensor is considerable more robust than the FD approach against sensor mismatch in low frequencies. The experimental results have shown that the KF u-sensor produced comparable velocity estimation as the Microflown sensor and notably better estimation than the FD method. All this indicates that the proposed u-sensor has the promise to serve as a costeffective particle velocity sensor. As the limitations of the present work, it is assumed in the proposed approach that the direction and the location of the target source are known a priori. In practical terms, model uncertainties due to location error and off-axis interfering sources may contribute to the velocity estimation errors. Investigations on these aspects are currently underway. Bai et al.: Particle velocity sensor

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ACKNOWLEDGMENT

The work was supported by the National Science Council (NSC) in Taiwan, under the project number NSC1002221-E-007-016-MY3. 1

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Bai et al.: Particle velocity sensor

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